Escape Velocity Calculation using Energy Conservation

[Void123]

I do not have a solution to a question I'm working on, so I just wanted to make sure the procedure was correct.

Homework Statement

I have a force acting on a particle in one dimension. It is given an initial speed v-0 and the problem wants to know what the maximum speed v-0 will be within the limits of energy conservation.

Homework Equations

Energy conservation and force (dependent on position).

The Attempt at a Solution

What I did was integrated to find the potential energy function. I then set v-final equal to zero and solved for v-0.

Is this approach correct?

 

[tiny-tim]

Welcome to PF!

Hi Void123! Welcome to PF!

(try using the X₂ tag just above the Reply box )

I have a force acting on a particle in one dimension. It is given an initial speed v-0 and the problem wants to know what the maximum speed v-0 will be within the limits of energy conservation.
…
What I did was integrated to find the potential energy function. I then set v-final equal to zero and solved for v-0.

Is this approach correct?

I don't follow exactly what the question is, and particularly what v_final has to do with it,

but generally you can use PE + KE = constant,

so maximum speed (which is what the question asks for) means maximum KE, and so means minimum PE.

 

[Void123]

Hi Void123! Welcome to PF!

(try using the X₂ tag just above the Reply box )


I don't follow exactly what the question is, and particularly what v_final has to do with it,

but generally you can use PE + KE = constant,

so maximum speed (which is what the question asks for) means maximum KE, and so means minimum PE.

I'm very sorry for not being specific enough. Actually, what the maximum speed was supposed to show was the speed it would have to reach before it can escape. The system I was dealing with was one of radial motion. Hence, if it reached an escape velocity the radius would go to infinity.

So, the initial speed (independent of the final) would have to be such that it would allow the particle to indefinitely escape the radial force holding it back.

I hope this clarifies everything

 

[ideasrule]

What I did was integrated to find the potential energy function. I then set v-final equal to zero and solved for v-0.

Is this approach correct?

Yes. If you use initial potential energy + initial kinetic energy = final potential energy (=0) + final kinetic energy (=0), you should have gotten the right answer.

 

[tiny-tim]

Hi Void123!

I'm very sorry for not being specific enough. Actually, what the maximum speed was supposed to show was the speed it would have to reach before it can escape. The system I was dealing with was one of radial motion. Hence, if it reached an escape velocity the radius would go to infinity.

ok, that's not a maximum, no wonder I was confused!

Let's look at the question again…

I have a force acting on a particle in one dimension. It is given an initial speed v-0 and the problem wants to know what the maximum speed v-0 will be within the limits of energy conservation.
…
What I did was integrated to find the potential energy function. I then set v-final equal to zero and solved for v-0.

Is this approach correct?

What you're actually asking is not for a maximum speed, but for the initial speed (v₀) which results in zero speed at infinity.

So yes, PE + KE = constant, and you want KE = 0 at infinity, so that means PE_∞ - PE₀ = KE₀

(btw, it doesn't have to be radial motion … with a conservative force, the path doesn't matter, and the escape speed will be the same, whatever the initial angle )